Bipartite Bell inequalities with three ternary-outcome measurements—from theory to experiments

Consider a Bell-type experiment involving two parties Alice and Bob, where each party is allowed to perform three ternary-outcome measurements. We label Alice's measurement setting (input) by x, Bob's by y and their corresponding measurement outcome (output) by a and b respectively. For ease of discussion, we follow the notation of [59] and refer to this as the $\{[3\;3\;3]\;[3\;3\;3]\}$ Bell scenario, where the number of entries in the first (second) square bracket is the number of input for Alice (Bob) while the actual value in the square brackets represents the number of output for that particular input.

The correlations between measurement outcomes observed in a Bell-type experiment in this scenario can be succinctly summarized using the vector $\vec{P}=\{P(a,b| x,y)\}{}_{x,y,a,b=0}^{2}$ of joint conditional probabilities. A correlation is said to be Bell-local [2, 3] if it admits the decomposition

$$\begin{eqnarray}&&P(a,b| x,y)\overset{{ \mathcal L }}{=}\displaystyle \sum _{\lambda }{P}_{\lambda }P(a| x,\lambda )P(b| y,\lambda )\end{eqnarray} \tag{ 1 }$$

for all $x,y,a,b$ with some fixed, normalized weights ${P}_{\lambda }\in [0,1]$, where we denote throughout by ${ \mathcal L }$ the set of Bell-local correlations. It turns out [60] that ${ \mathcal L }$ is a convex polytope [61], and thus can be described by a convex mixture of a finite number of (deterministic) extremal probability vectors satisfying $P(a| x,\lambda )=0,1$ and $P(b| y,\lambda )=0,1$. Equivalently, a convex polytope can be fully characterized by the intersection of a finite number of half-spaces [61]. Following [30], we refer to the minimal set of such half-spaces as facet-defining Bell inequalities, or simply as facets. A generic Bell inequality, i.e., one involving some linear combination of joint conditional probabilities, for the $\{[3\;3\;3]\;[3\;3\;3]\}$ scenario reads as

$$\begin{eqnarray}&&\displaystyle \sum _{x,y,a,b=0}^{2}{\alpha }_{{ab}}^{{xy}}P(a,b| x,y)\overset{{ \mathcal L }}{\leqslant }{{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}(\{{\alpha }_{{ab}}^{{xy}}\})\end{eqnarray} \tag{ 2 }$$

and is, however, not necessarily a facet. Here, we write explicitly the dependence of ${{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}$ on $\{{\alpha }_{{ab}}^{{xy}}\}$ to remind that the upper bound attainable by $\vec{P}\in { \mathcal L }$ is a function of the real coefficients $\{{\alpha }_{{ab}}^{{xy}}\}$.

As was first shown by Bell, the set of correlation vectors $\vec{P}$ allowed in quantum theory (denoted by ${ \mathcal Q }$) is a strict superset of ${ \mathcal L }$. Formally, quantum correlations take the form of

$$\begin{eqnarray}&&P(a,b| x,y)\overset{{ \mathcal Q }}{=}\mathrm{tr}(\rho \;{M}_{a| x}\otimes {M}_{b| y}),\end{eqnarray} \tag{ 3a }$$

where ρ is a density matrix and ${M}_{a| x}$ and ${M}_{b| y}$ are the positive-operator-valued measure (POVM) elements associated with Alice's and Bob's local measurements, i.e., they satisfy

$$\begin{eqnarray}&&{M}_{a| x}\geqslant 0,\;{M}_{b| y}\geqslant 0,\;\displaystyle \sum _{a}{M}_{a| x}={{\mathbb{1}}}_{A},\;\displaystyle \sum _{b}{M}_{b| y}={{\mathbb{1}}}_{B}\end{eqnarray} \tag{ 3b }$$

for all $x,y,a,b$, with ${{\mathbb{1}}}_{A}({{\mathbb{1}}}_{B})$ being the identity operator acting on Alice's (Bob's) Hilbert space.

It is easy to see that the marginal distributions of the quantum correlation, $P(a| x,y)$ and $P(b| x,y)$, are independent of the input of the other party, i.e., they satisfy the so-called non-signaling (NS) conditions [62, 63]

$$\begin{eqnarray}P(a| x,y) & & \equiv \displaystyle \sum _{b}P(a,b| x,y)\overset{{ \mathcal N }{ \mathcal S }}{=}P(a| x),P(b| x,y) & & \equiv \displaystyle \sum _{a}P(a,b| x,y)\overset{{ \mathcal N }{ \mathcal S }}{=}P(b| y).\end{eqnarray} \tag{ 4 }$$

Note that if these conditions are violated independent of spatial separation, then Alice can communicate superluminally the value of x to Bob by remotely varying the marginal distribution observed by Bob through her choice of x. Interestingly, the set of correlations satisfying equation (4), which we shall denote by ${ \mathcal N }{ \mathcal S }$ is actually a strict superset of ${ \mathcal Q }$ (see, for instance, [3, 64] and references therein).

Due to the non-signaling nature of $\vec{P}\in \{{ \mathcal L },{ \mathcal Q }\}$, instead of specifying ${3}^{4}+1=82$ real coefficients ${\rm{(}}{\alpha }_{{ab}}^{{xy}}$ and ${{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}(\{{\alpha }_{{ab}}^{{xy}}\}){\rm{)}}$ in defining a Bell inequality, one can employ a more compact representation due to Collins and Gisin [36], which requires only the specification of ${(2\times 3+1)}^{2}=49$ parameters. Explicitly, the Collins–Gisin representation of a Bell inequality in this scenario reads as

$$\begin{eqnarray}\vec{\beta }\cdot \vec{P} & = & \displaystyle \sum _{x=0}^{2}\displaystyle \sum _{a=0}^{1}{\beta }_{A,a}^{x}P(a| x)+\displaystyle \sum _{y=0}^{2}\displaystyle \sum _{b=0}^{1}{\beta }_{B,b}^{y}P(b| y)+\displaystyle \sum _{x,y=0}^{2}\displaystyle \sum _{a,b=0}^{1}{\beta }_{{ab}}^{{xy}}P(a,b| x,y)\overset{{ \mathcal L }}{\leqslant }{{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}(\vec{\beta }),\end{eqnarray} \tag{ 5 }$$

where $\vec{\beta }$ is a vector with entries given by the Bell coefficients ${\beta }_{A,a}^{x}$, ${\beta }_{B,b}^{y}$, ${\beta }_{{ab}}^{{xy}}$ appearing in equation (5), while $\vec{P}$ is now the corresponding vector of marginal and joint conditional probability distributions. This particular way of writing a Bell inequality admits the compact tabular representation

$$\begin{eqnarray}\left(\begin{array}{rcccccc} & {\beta }_{B,0}^{0} & {\beta }_{B,1}^{0} & {\beta }_{B,0}^{1} & {\beta }_{B,1}^{1} & {\beta }_{B,0}^{2} & {\beta }_{B,1}^{2}\\ {\beta }_{A,0}^{0} & {\beta }_{00}^{00} & {\beta }_{01}^{00} & {\beta }_{00}^{01} & {\beta }_{01}^{01} & {\beta }_{00}^{02} & {\beta }_{01}^{02}\\ {\beta }_{A,1}^{0} & {\beta }_{10}^{00} & {\beta }_{11}^{00} & {\beta }_{10}^{01} & {\beta }_{11}^{01} & {\beta }_{10}^{02} & {\beta }_{11}^{02}\\ {\beta }_{A,0}^{1} & {\beta }_{00}^{10} & {\beta }_{01}^{10} & {\beta }_{00}^{11} & {\beta }_{01}^{11} & {\beta }_{00}^{12} & {\beta }_{01}^{12}\\ {\beta }_{A,1}^{1} & {\beta }_{10}^{10} & {\beta }_{11}^{10} & {\beta }_{10}^{11} & {\beta }_{11}^{11} & {\beta }_{10}^{12} & {\beta }_{11}^{12}\\ {\beta }_{A,0}^{2} & {\beta }_{00}^{20} & {\beta }_{01}^{20} & {\beta }_{00}^{21} & {\beta }_{01}^{21} & {\beta }_{00}^{22} & {\beta }_{01}^{22}\\ {\beta }_{A,1}^{2} & {\beta }_{10}^{20} & {\beta }_{11}^{20} & {\beta }_{10}^{21} & {\beta }_{11}^{21} & {\beta }_{10}^{22} & {\beta }_{11}^{22}\end{array}\right)\overset{{ \mathcal L }}{\leqslant }{{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}(\vec{\beta }),\end{eqnarray} \tag{ 6 }$$

so that the left-most column are the Bell coefficients associated with Alice's marginal probabilities, the top row gives the Bell coefficients for Bob's marginal probabilities, while each xth block row and yth block column gives the Bell coefficients for the joint distribution of the input combination (x, y).

With the judicious choice of an entangled quantum state ρ and local measurements described by ${M}_{a| x}$ and ${M}_{b| y}$, the resulting quantum correlation ${\vec{P}}_{{ \mathcal Q }}$ may lead to the violation of a Bell inequality. Given the identities of equation (4), it is clear that a Bell inequality can be written in infinitely many different forms. Due to this arbitrariness, the difference between the corresponding quantum value ${{ \mathcal S }}^{{ \mathcal Q }}(\vec{\beta },\rho ,\{{M}_{a| x},{M}_{b| y}\})$ and the local bound ${{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}(\vec{\beta })$ is thus not a good figure of merit for comparing different Bell inequalities. Rather, a commonly adopted measure that is unaffected by such an arbitrariness is given by the extent to which the correlation ${\vec{P}}_{{ \mathcal Q }}$ can tolerate white noise before it stops violating the given Bell inequality.

Formally, let us denote the uniform probability distribution (white noise) for the Bell scenario $\{[3\;3\;3]\;[3\;3\;3]\}$ by ${\vec{P}}_{{\mathbb{1}}}$, i.e., ${P}_{{\mathbb{1}}}(a,b| x,y)=\frac{1}{{3}^{2}}$ for all $a,b,x,y$. Then for any given (nonlocal) quantum correlation ${\vec{P}}_{{ \mathcal Q }}$ that violates a Bell inequality ${ \mathcal I }$, let us consider the convex mixture

$$\begin{eqnarray}&&{\vec{P}}_{v}=v{\vec{P}}_{{ \mathcal Q }}+(1-v){\vec{P}}_{{\mathbb{1}}},\quad 0\leqslant v\leqslant 1.\end{eqnarray} \tag{ 7 }$$

Clearly, since ${\vec{P}}_{{ \mathcal Q }}\not\hspace{-2pt}{\in }{ \mathcal L }$, ${\vec{P}}_{{\mathbb{1}}}\in { \mathcal L }$ and ${ \mathcal L }$ is convex, as we decrease the value of v from 1, there is some critical value ${v}_{{\rm{Cr}}}^{{ \mathcal I }}$ such that for all $v\in [0,{v}_{{\rm{Cr}}}^{{ \mathcal I }}]$, ${\vec{P}}_{v}$ does not violate ${ \mathcal I }$. This critical value ${v}_{{\rm{Cr}}}^{{ \mathcal I }}$—which does not depend on how ${ \mathcal I }$ is represented—is commonly referred to as the (white-noise) visibility of ${\vec{P}}_{{ \mathcal Q }}$ with respect to ${ \mathcal I }$. More generally, as we decrease the value of v from 1, due to the convexity of ${ \mathcal L }$, there is some critical value ${v}_{{\rm{Cr}}}$ such that for all $v\in [0,{v}_{{\rm{Cr}}}]$, ${\vec{P}}_{v}\in { \mathcal L }$. Hereafter, we refer to this critical value as the (white-noise) visibility of ${\vec{P}}_{{ \mathcal Q }}$ with respect to ${ \mathcal L }$. It is worth noting that for any given nonlocal ${\vec{P}}_{{ \mathcal Q }}$, ${v}_{{\rm{Cr}}}$ can be efficiently computed using linear programing (see appendix A for details).

In the event when ${\vec{P}}_{{ \mathcal Q }}$ arises from measuring only rank-1 projectors on an entangled two-qutrit state ρ, ${v}_{{\rm{Cr}}}^{{ \mathcal I }}$ defined above also coincides with the infimum of ν in

$$\begin{eqnarray}&&{\rho }_{\nu }=\nu \rho +(1-\nu )\displaystyle \frac{{\mathbb{1}}}{{d}^{2}},\quad 0\leqslant \nu \leqslant 1,\end{eqnarray} \tag{ 8 }$$

before ${\rho }_{\nu }$ stops violating ${ \mathcal I }$ for the given measurements $\{{M}_{a| x},{M}_{b| y}\};$ in equation (8), d is the local Hilbert space dimension of ρ, e.g., d = 3 in the case of a qutrit. Hereafter, we refer to the infimum of ν in equation (8) for the general scenario, i.e., when ρ is not necessarily a two-qutrit state and/or $\{{M}_{a| x},{M}_{b| y}\}$ are not necessarily rank-1 projectors as the state visibility ${\nu }_{{\rm{Cr}}}^{{ \mathcal I }}$ of ρ with respect to ${ \mathcal I }$. As it stands, this visibility depends not only on ρ and ${ \mathcal I }$, but also on the choice of POVM elements $\{{M}_{a| x},{M}_{b| y}\}$. The visibilities ${\nu }_{{\rm{Cr}}}^{I}$ and ${v}_{{\rm{Cr}}}^{I}$ will be our main figures of merit in comparing the different Bell inequalities presented in the next section.

Let us now briefly recapitulate the state of the art of various Bell scenarios. For $\{[2\;2]\;[2\;2]\}$, $\{[3\;3]\;[3\;3]\}$, $\{[2\;2\;2]\;[2\;2\;2]\}$, $\{[2\;2\;2]\;[2\;2\;2\;2]\}$, $\{[2\;3]\;[2\;2\;2]\}$ and $\{[2\;2]\;[2\;2]\;[2\;2]\}$, the complete list of facet-defining Bell inequalities has been obtained with the help of standard polytope softwares, see [28, 36, 65]. Generalizing the results of [28, 36], Pironio [65] recently showed that there is a large family of Bell scenarios where the only non-trivial facets are either the CHSH inequality or their liftings [66]. Beyond this, only partial lists of facet-defining Bell inequalities are known. Specifically, those for $\{[2\;2\;2\;2]\;[2\;2\;2\;2]\}$ and $\{[2\;2\;2\;2\;2]\;[2\;2\;2\;2\;2]\}$ can be found in [32, 33] (the recent work by Deta and Sikirić [67], however, suggests that the known list of 175 facets [68] for $\{[2\;2\;2\;2]\;[2\;2\;2\;2]\}$ is complete).

In this paper, we shall restrict our attention to the Bell scenario $\{[3\;3\;3]\;[3\;3\;3]\}$ where, to our knowledge, the only known (non-lifted) nontrivial facet is the one presented in [36]. To get a better idea of what quantum entanglement has to offer in this scenario, we shall first generate some novel facets for this scenario. While a few techniques [30, 31, 33] are known in the literature for generating facets for ${ \mathcal L }$, here we adopt a different approach—based on linear programming—which allows us to obtain nontrivial facets that can be violated by quantum theory with some nontrivial ${v}_{{\rm{Cr}}}$. It is worth noting that the search for Bell inequality using linear programming has also been considered [37] in the context of minimizing the detection efficiency requirement in a loophole free Bell test.

Let us now recall from [69] the following Bell inequality (first introduced in [40])

$$\begin{eqnarray}&&{I}_{3}^{+}\;:{{ \mathcal S }}_{0}=\displaystyle \frac{1}{9}\displaystyle \sum _{x,y,a,b=0}^{2}{\delta }_{{xy}+a+b}\;P(a,b| x,y)\overset{{ \mathcal L }}{\leqslant }\displaystyle \frac{2}{3},\end{eqnarray} \tag{ 9 }$$

where ${\delta }_{f}$ is a short-hand for the Kronecker delta ${\delta }_{f\mathrm{mod}\mathrm{3,0}}$. This inequality was rediscovered (in a different form) in [70] and has been discussed as a specific case of a family of generalization of the CHSH Bell inequalities [69, 71–75]. Maximal quantum violation ($\approx 0.7124$) of this inequality can be achieved by (locally) performing mutually unbiased measurements on the maximally entangled two-qutrit state [69] (see also [70]), i.e.,

$$\begin{eqnarray}&&| {{\rm{\Phi }}}_{3}^{+}\rangle =\displaystyle \frac{1}{\sqrt{3}}\displaystyle \sum _{i=0}^{2}{| i\rangle }_{{\rm{A}}}{| i\rangle }_{{\rm{B}}},\end{eqnarray} \tag{ 10 }$$

where $\{{| i\rangle }_{{\rm{A}}}\}$ are orthonormal basis vectors for Alice's (qutrit) Hilbert space; likewise for $\{{| i\rangle }_{{\rm{B}}}\}$. This feature of ${I}_{3}^{+}$ naturally suggests that it may be used for the self-testing of $| {{\rm{\Phi }}}_{3}^{+}\rangle$. Indeed, numerical optimization using the tools of [23] shows that when one approaches the maximal quantum violation of ${I}_{3}^{+}$, the underlying quantum state must also have a negativity that approaches 1, which is exactly the negativity of $| {{\rm{\Phi }}}_{3}^{+}\rangle$. However, the corresponding quantum correlation ${\vec{P}}_{{ \mathcal Q }}^{{I}_{3}^{+}}$—as was shown in [69]—exhibits only a white-noise visibility ${v}_{{\rm{Cr}}}^{{I}_{3}^{+}}$ of 87.94%.

In addition, while being a natural generalization of the CHSH Bell inequality to three ternary-outcome measurements, the Bell inequality ${I}_{3}^{+}$ is provably [69] not facet-defining. Thus, it can be written as a convex combination of facets [61] such that at least one of which offers at least as good (if not better) white-noise visibility for the correlation ${\vec{P}}_{{ \mathcal Q }}^{{I}_{3}^{+}}$. In particular, if this facet is also maximally violated by $| {{\rm{\Phi }}}_{3}^{+}\rangle$, one will have obtained a more-promising candidate for the self-testing of a maximally entangled two-qutrit state.

How then do we look for the constituent facet-defining Bell inequalities which give rise to inequality ${I}_{3}^{+}$? It turns out that in computing the critical visibility ${v}_{{\rm{Cr}}}$ of any given nonlocal correlation ${\vec{P}}_{{ \mathcal Q }}$, the (dual of the) linear program also outputs a Bell inequality ${ \mathcal I }$ such that ${\vec{P}}_{{ \mathcal Q }}$ has a visibility of ${v}_{{\rm{Cr}}}$ with respect to ${ \mathcal I }$ (see appendix A for details). For a generic nonlocal correlation ${\vec{P}}_{{ \mathcal Q }}$, as we decrease the weight v, the convex combination given in equation (7) enters the local polytope ${ \mathcal L }$ via one of its facets and the Bell inequality outputted by the linear program is thus also facet-defining.

Indeed, by solving the linear program of equation (A1) using the correlation ${\vec{P}}_{{ \mathcal Q }}^{{I}_{3}^{+}}$, we obtain the 16th facet-defining Bell inequality listed in table 1 with the same critical visibility offered by ${I}_{3}^{+}$. More generally, by maximizing the quantum violation of ${I}_{3}^{+}$ using two-qutrit entangled states of the form

$$\begin{eqnarray}&&| \psi (\gamma ,{\gamma }^{\prime })\rangle =\displaystyle \frac{{| 0\rangle }_{{\rm{A}}}{| 0\rangle }_{{\rm{B}}}+\gamma {| 1\rangle }_{{\rm{A}}}{| 1\rangle }_{{\rm{B}}}+{\gamma }^{\prime }{| 2\rangle }_{{\rm{A}}}{| 2\rangle }_{{\rm{B}}}}{\sqrt{1+{\gamma }^{2}+{\gamma }^{\prime 2}}},\end{eqnarray} \tag{ 11 }$$

with $\gamma \in [0,1]$ and ${\gamma }^{\prime }=1$, we obtain nonlocal correlations that exhibit ${v}_{{\rm{Cr}}}^{{I}_{3}^{+}}\in [0.8794,0.9683]$. Then, using these correlations as input to the linear program given in equation (A1), we obtain another 15 facet-defining Bell inequalities, listed as inequality 1–15 in table 1. Interestingly, among these 16 facets, only 12 are genuinely novel facets for the Bell scenario $\{[3\;3\;3]\;[3\;3\;3]\}$ while inequality 12, 13, 15, and 16—as can be verified using the online platform [76] (see also [77])—are reducible, respectively, to the simpler⁴ Bell scenario $\{[3\;3\;2]\;[3\;3\;2]\}$ and $\{[3\;2]\;[2\;2\;2]\}$. Finally, we find two other facets by using the (post-processed) experimentally observed correlations (see section 5) as input to the linear program (equation (A1)). Hereafter, we denote all these inequalities by ${{ \mathcal I }}_{n}^{{\rm{max}}}$ with $n\in 1,2,\ldots ,18$, i.e.,

$$\begin{eqnarray}&&{{ \mathcal I }}_{n}^{{\rm{max}}}\;:{{ \mathcal S }}_{n}\equiv {\vec{\beta }}_{n}\cdot \vec{P}\overset{{ \mathcal L }}{\leqslant }{{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}({\vec{\beta }}_{n}).\end{eqnarray} \tag{ 12 }$$

Table 1.  Coefficients of facet-defining Bell inequalities ${{ \mathcal I }}_{n}^{{\rm{max}}}\;:{{ \mathcal S }}_{n}=\vec{\beta }\cdot \vec{P}\leqslant {{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}$ found by solving the linear program given in equation (A1). The left-most column gives the inequality number, i.e., n in ${{ \mathcal I }}_{n}^{{\rm{max}}}$, whereas the second column gives the local bound ${{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}$, i.e., the maximum value of the Bell expression allowed by all $\vec{P}\in { \mathcal L }$, see equation (5). The coefficients $\vec{\beta }$ of the Bell inequalities are sorted according to the order in which they appear in the table of equation (6), i.e., first from the top row to the bottom row, then from the left-most column to the right-most column: ${\beta }_{A,0}^{0}$, ${\beta }_{A,1}^{0}$, ${\beta }_{A,0}^{1}$, ..., ${\beta }_{11}^{22}$.

n	${{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}$	${\vec{\beta }}_{n}$
1	2	0	1	0	1	−1	0	0	1	0	1	−2	1	−1	1	0	0	0	−1	0	−1	−1	1	0	0	1	0	−1	0	−2	−1	1	0	−1	1	0	1	0	−1	−1	0	1	1	−1	−1	−1	0	1	−1
2	2	−1	1	0	1	−1	0	0	2	0	1	−2	1	−2	1	1	0	0	−1	0	−1	−1	1	0	0	1	0	0	0	−1	−1	1	0	−1	1	0	1	0	−1	−1	0	1	1	−1	−1	−1	0	1	−1
3	2	0	1	0	1	−1	0	0	1	0	1	−2	1	−2	1	0	0	0	−1	0	−1	−1	1	0	0	1	0	−1	0	−1	−1	1	0	0	1	0	1	0	−1	−1	0	1	1	−1	−1	−2	0	1	−1
4	2	0	1	−1	0	0	1	−1	2	1	1	−1	1	−2	1	0	0	0	−1	0	−1	0	1	0	0	1	−1	−2	1	−1	−1	1	0	−1	0	−1	1	0	0	0	0	1	0	−2	−1	0	1	1	−1
5	2	0	1	0	1	−1	0	0	1	0	1	−2	1	−2	1	0	0	0	−1	0	−1	−1	1	0	0	1	0	0	0	−1	−1	1	0	0	1	0	1	0	−1	−2	0	1	1	−1	−1	−2	0	1	−1
6	2	−1	1	0	1	−1	0	0	2	0	1	−2	1	−2	1	1	0	0	−1	0	−1	−1	1	0	0	1	0	0	0	−1	−1	1	0	0	1	0	1	0	−1	−1	0	1	1	−1	−1	−2	0	1	−1
7	2	0	1	0	1	−1	0	0	1	0	1	−2	1	−2	1	0	0	0	−1	0	−1	−1	1	0	0	1	0	−1	0	−1	−1	1	−1	0	1	0	1	0	−1	−1	0	1	1	−1	−1	−2	0	1	0
8	2	−1	1	−1	0	0	1	0	2	0	1	−2	1	−2	1	1	0	0	−1	0	−1	0	1	0	0	1	−1	−1	1	−1	−1	1	0	−1	0	−1	1	0	0	0	0	1	0	−1	−1	−1	1	1	−1
9	2	0	1	−1	0	0	1	0	1	0	1	−2	1	−2	1	0	0	0	−1	0	−1	0	1	0	0	1	−1	−2	1	−1	−1	1	0	−2	0	−1	1	0	0	0	0	1	0	−1	−1	0	1	1	−1
10	1	0	1	−1	0	−1	0	0	1	0	1	−2	1	−1	1	0	−1	0	−1	0	−1	−1	1	0	0	1	0	0	0	−2	−1	1	−1	0	1	−1	1	0	0	−1	0	1	0	−1	−1	−1	1	1	0
11	2	0	1	0	1	−1	0	0	1	0	1	−2	1	−1	1	0	0	0	−1	0	−1	−1	1	0	0	1	0	−1	0	−2	−1	1	0	0	1	0	1	0	−1	-1	0	1	1	−1	-1	−2	0	1	−1
12	2	0	1	0	1	−1	0	1	1	0	0	−2	1	−1	1	1	0	0	−1	0	−1	0	0	0	0	0	0	0	1	−2	−1	0	−1	−1	1	−1	1	0	0	0	0	1	0	−1	−1	0	1	1	−1
13	2	−1	1	−1	0	0	1	0	2	0	1	−2	1	−2	1	1	0	0	−1	0	−1	0	1	0	0	1	−1	−1	1	−1	−1	1	0	−2	0	−1	1	0	0	0	0	1	0	−1	−1	0	1	1	−1
14	2	0	1	0	1	−1	0	0	1	0	1	−2	1	−1	1	0	0	0	−1	0	−1	−1	1	0	0	1	0	0	0	−2	−1	1	0	0	1	0	1	0	−1	−2	0	1	1	−1	−1	−2	0	1	−1
15	1	−2	1	0	0	0	−1	0	2	−1	0	−1	1	−1	1	1	−1	0	−1	0	−1	−1	1	−1	0	2	−1	1	0	0	−1	0	1	0	1	0	1	0	0	−1	−1	1	0	0	0	0	0	0	0
16	1	0	1	0	0	0	0	−1	1	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	−1	−1	1	0	0	0	1	0	−1	−1	0	0	0	1	0	−1	-1	0	0	0
17	1	1	1	0	0	0	−1	0	0	0	0	0	0	0	1	−1	−1	−1	0	−1	−1	0	0	0	−1	0	0	1	0	−1	−1	−1	0	1	1	0	0	0	0	0	0	0	0	−1	-1	1	0	0	1
18	1	1	1	0	0	0	−1	0	0	0	0	0	0	0	1	−1	−1	0	0	−1	−1	0	0	0	−1	0	0	1	0	−1	−1	−1	0	1	1	0	0	0	0	0	0	0	−1	−1	−1	1	0	1	2
19	1	0	1	0	0	0	0	0	0	−1	0	−1	1	0	1	−1	−1	−1	−1	0	−1	0	0	−1	1	0	−1	0	0	−1	−1	0	−1	0	1	0	1	0	−1	0	−1	0	0	0	−1	0	1	0	0

Evidently, as can be seen in table 2, all the 18 Bell inequalities presented can be violated quantum mechanically with a visibility ${v}_{{\rm{Cr}}}={v}_{{\rm{max}}}$ much better than the starting visibility ${v}_{{\rm{Cr}}}^{{I}_{3}^{+}}\in [0.8794,0.9683]$. Moreover, except for ${{ \mathcal I }}_{16}^{{\rm{max}}}$ and ${{ \mathcal I }}_{17}^{{\rm{max}}}$—both of which can be reduced to a simpler Bell scenario involving a combination of binary-outcome and ternary-outcome measurements—the maximal quantum violation⁵ of the rest can already be achieved using two-qubit entangled states, including the maximally entangled two-qubit state $| {{\rm{\Phi }}}^{+}\rangle =| \psi (1,0)\rangle$, see equation (11). These inequalities therefore serve, to our knowledge, the first examples of non-lifted facet-defining Bell inequalities whose maximal quantum violation is attainable using a Hilbert space dimension smaller than the number of possible outcomes involved.

Table 2.  Summary of some of the properties of the Bell inequalities ${{ \mathcal I }}_{n}^{{\rm{max}}}\;:{{ \mathcal S }}_{n}\leqslant {{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}$ (3rd-7th column) and ${{ \mathcal I }}_{n}^{{\rm{min}}}\;:{{ \mathcal S }}_{n}\geqslant {{ \mathcal S }}_{{\rm{min}}}^{{ \mathcal L }}$ (8th-12th column) for the Bell expression ${{ \mathcal S }}_{n}$ presented in table 1. The second column gives the simplest Bell scenario to which ${{ \mathcal S }}_{n}$ can be reduced. Within each block column, we list the maximum (minimum) local value ${{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}$ (${{ \mathcal S }}_{{\rm{min}}}^{{ \mathcal L }}$), the maximum (minimum) quantum value ${{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal Q }}$, the (Schmidt coefficients of the) state $| {\psi }_{{\rm{max}}}\rangle$ ($| {\psi }_{{\rm{min}}}\rangle$) found to achieve ${{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal Q }}$ (${{ \mathcal S }}_{{\rm{min}}}^{{ \mathcal Q }}$), the corresponding state visibility ${\nu }_{{\rm{Cr}}}^{I}$ and white-noise visibility ${v}_{{\rm{Cr}}}^{I}$. The dimension spanned by the set of local extreme points saturating the minimal local value ${{ \mathcal S }}_{{\rm{min}}}^{{ \mathcal L }}$ is given in the last column. Entries marked with † means that within the numerical precision of the solver, the corresponding Bell inequality ${{ \mathcal I }}_{n}^{{\rm{min}}}$ is provably satisfied by quantum theory. For the entry marked with ‡, there remains a gap of the order of $2.7\times {10}^{-3}$ between the ${{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal Q }}$ presented and the best upper bound that we obtained from local level 2 of the hierarchy considered in [23].

n	Scenario	${{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal L }}$	${{ \mathcal S }}_{{\rm{max}}}^{{ \mathcal Q }}$	$| {\psi }_{{\rm{max}}}\rangle$	${\nu }_{{\rm{max}}}$	${v}_{{\rm{max}}}$	${{ \mathcal S }}_{{\rm{min}}}^{{ \mathcal L }}$	${{ \mathcal S }}_{{\rm{min}}}^{{ \mathcal Q }}$	$| {\psi }_{{\rm{min}}}\rangle$	${\nu }_{{\rm{min}}}$	${v}_{{\rm{min}}}$	dmin
1	$\{[3\;3\;3]\;[3\;3\;3]\}$	2	2.6972	$| {{\rm{\Phi }}}^{+}\rangle$	0.7819	0.7415	−4${}^{\dagger }$	−4.0000	—	—	—	1
2	$\{[3\;3\;3]\;[3\;3\;3]\}$	2	2.6712	$| {{\rm{\Phi }}}^{+}\rangle$	0.7884	0.7487	−4${}^{\dagger }$	−4.0000	—	—	—	1
3	$\{[3\;3\;3]\;[3\;3\;3]\}$	2	2.6586	[0.7278; 0.6857]	0.7915	0.7523	−5${}^{\dagger }$	−5.0000	—	—	—	2
4	$\{[3\;3\;3]\;[3\;3\;3]\}$	2	2.6586	[0.7278; 0.6857]	0.7915	0.7523	−5${}^{\dagger }$	−5.0000	—	—	—	2
5	$\{[3\;3\;3]\;[3\;3\;3]\}$	2	2.6488	[0.7129; 0.7013]	0.7940	0.7551	−4${}^{\dagger }$	−4.0000	—	—	—	4
6	$\{[3\;3\;3]\;[3\;3\;3]\}$	2	2.6577	[0.7222; 0.6917]	0.7917	0.7525	−4${}^{\dagger }$	−4.0000	—	—	—	4
7	$\{[3\;3\;3]\;[3\;3\;3]\}$	2	2.6577	[0.7222; 0.6917]	0.7917	0.7525	−4${}^{\dagger }$	−4.0000	—	—	—	5
8	$\{[3\;3\;3]\;[3\;3\;3]\}$	2	2.6577	[0.7222; 0.6917]	0.7917	0.7525	−4	−4.0010	[0.7925; 0.5893; 0.1568]	0.9997	0.9997	5
9	$\{[3\;3\;3]\;[3\;3\;3]\}$	2	2.6720	[0.7150; 0.6992]	0.7881	0.7485	−4	−4.0171	[0.7468; 0.6175; 0.2470]	0.9941	0.9941	6
10	$\{[3\;3\;3]\;[3\;3\;3]\}$	1	1.6720	[0.7150; 0.6992]	0.7881	0.7485	−5	−5.0171	[0.7468; 0.6175; 0.2470]	0.9941	0.9941	7
11	$\{[3\;3\;3]\;[3\;3\;3]\}$	2	2.6955	[0.7230; 0.6908]	0.7824	0.7420	−4	−4.0138	[0.7932; 0.5571; 0.2457]	0.9952	0.9952	8
12	$\{[3\;3\;2]\;[3\;3\;2]\}$	2	2.5820	[0.7258; 0.6879]	0.7746	0.7277	−3	−3.0005	[0.8957; 0.4328; 0.1023]	0.9998	0.9998	8
13	$\{[3\;3\;2]\;[3\;3\;2]\}$	2	2.6712	$| {{\rm{\Phi }}}^{+}\rangle$	0.7884	0.7487	−3	−3.6712	$| {{\rm{\Phi }}}^{+}\rangle$	0.7884	0.8172	26
14	$\{[3\;3\;3]\;[3\;3\;3]\}$	2	2.6972	$| {{\rm{\Phi }}}^{+}\rangle$	0.7819	0.7415	−3	−3.6972	$| {{\rm{\Phi }}}^{+}\rangle$	0.7819	0.8114	27
15	$\{[3\;3\;2]\;[3\;3\;2]\}$	1	1.5923	$| {{\rm{\Phi }}}^{+}\rangle$	0.7715	0.7242	−3	−3.5923	$| {{\rm{\Phi }}}^{+}\rangle$	0.7715	0.8050	29
16	$\{[3\;2]\;[2\;2\;2]\}$	1	1.2532	[0.6608; 0.5307; 0.5307]	0.7247	0.7247	−1	−1.0328	[0.6773; 0.6769; 0.2882]	0.9682	0.9682	30
17	$\{[3\;2\;2]\;[3\;2\;2]\}$	1	1.3090	[0.6388; 0.5860; 0.4985]	0.7639	0.7639	−3${}^{\dagger }$	−3.0000	—	—	—	4
18	$\{[3\;2\;2]\;[3\;2\;2]\}$	1	1.4142	$| {{\rm{\Phi }}}^{+}\rangle$	0.7071	0.7071	−2${}^{\dagger }$	−2.0000	—	—	—	16
19	$\{[3\;3\;3]\;[3\;3\;3]\}$	1	${1.3782}^{ddagger}$	[0.6069; 0.6064; 0.5137]	0.7925	0.7925	−3	−3.2071	$| {{\rm{\Phi }}}^{+}\rangle$	0.7071	0.9250	13

Given the above observation, a natural question that one may now ask is whether a genuine POVM (i.e., measurement involving non-projective operators) is 'needed' to achieve these maximal quantum violations. Of course, if there is no restriction in the Hilbert space dimension, the maximal quantum violation of a Bell inequality can always be achieved by performing only projective measurements in a sufficiently large Hilbert space (see, for instance, [82, 83]). The above question is thus relevant only if we restrict ourselves to the minimal Hilbert space dimension where the maximal quantum violation of a given Bell inequality is known to be achievable.

Interestingly, we have found that for all but one of these 'qubit' inequalities, it is already sufficient to allow the trivial projector ${{\bf{0}}}_{2}$, i.e., the 2 × 2 zero matrix in the three-outcome measurement in order to recover the maximal quantum violation. The only exception to these is ${{ \mathcal I }}_{12}^{{\rm{max}}}$, where we could show—by considering all possible combinations of trivial-projector assignments and using a suitable modification of the SDP of [79] as well as the SDP discussed in [81]—that to achieve maximal quantum violation of ${{ \mathcal I }}_{12}^{{\rm{max}}}$ using a quantum state of the lowest possible dimension necessarily requires non-projective measurements. In contrast with previous example [84] where a genuine POVM was found to be relevant, our example here actually makes use of a facet-defining Bell inequality, thus answering a stronger form of the opened problem (fundamental question no. 10) posed by Gisin in [85]. Moreover, our example is also considerably more noise-resistant than that of [84], as we can tolerate⁶ between 6.66%–17.54% of white noise, before losing the advantage of genuine POVM over projective measurements using ${{ \mathcal I }}_{12}^{{\rm{max}}}$. For an explicit set of genuine POVM leading to the maximal quantum violation of ${{ \mathcal I }}_{12}^{{\rm{max}}}$, see appendix B.

For completeness, we also analyze the quantum violation of the only $\{[3\;3\;3]\;[3\;3\;3]\}$ facet known in the literaute [36], which we list as ${{ \mathcal I }}_{19}^{{\rm{max}}}$ (and cast in a more symmetrical form) in table 1 . Likewise, we also carry out detailed analysis of the Bell inequality that result from the minimum (see equation (5)) of the Bell expression given in table 1 over all correlations in ${ \mathcal L }$, i.e.,

$$\begin{eqnarray}&&{{ \mathcal I }}_{n}^{{\rm{min}}}\;:{S}_{n}={\vec{\beta }}_{n}\cdot \vec{P}\overset{{ \mathcal L }}{\geqslant }{{ \mathcal S }}_{{\rm{min}}}^{{ \mathcal L }}({\vec{\beta }}_{n}).\end{eqnarray} \tag{ 13 }$$

The results of these investigations are also summarized in table 2. Not surprisingly, all the Bell inequalities ${{ \mathcal I }}_{n}^{{\rm{min}}}$ with $n\in \{1,2,\ldots ,19\}$ are not facet-defining. Moreover, from the numerical upper bound obtained from [79], nine of these inequalities cannot even be violated quantum mechanically, while five others can be violated with rather poor visibility ${v}_{{\rm{Cr}}}^{I}$ using some partially entangled two-qutrit states. The remaining four inequalities ${{ \mathcal I }}_{13}^{{\rm{min}}}$, ${{ \mathcal I }}_{14}^{{\rm{min}}}$, ${{ \mathcal I }}_{15}^{{\rm{min}}}$, and ${{ \mathcal I }}_{19}^{{\rm{min}}}$ turn out to be maximally violated by $| {{\rm{\Phi }}}^{+}\rangle$. Moreover, the maximal quantum violation of ${{ \mathcal I }}_{n}^{{\rm{min}}}$ with $n\in \{13,14,15\}$ turns out to give exactly the same state visibility as their counterpart ${{ \mathcal I }}_{n}^{{\rm{max}}}$.

For ${{ \mathcal I }}_{19}^{{\rm{max}}}$, the best quantum violation that we have found is attained using a partially entangled two-qutrit state. However, since we cannot close the gap between this quantum violation with the upper bound coming from the NPA hierarchy [79] (or the hierarchy considered in [23]), we cannot yet conclude that the presented value indeed represents the maximal quantum violation of ${{ \mathcal I }}_{19}^{{\rm{max}}}$. Finally, let us note from table 2 that the Bell inequalities listed within each of the three sets $\{{{ \mathcal I }}_{3}^{{\rm{max}}},{{ \mathcal I }}_{4}^{{\rm{max}}}\}$, $\{{{ \mathcal I }}_{6}^{{\rm{max}}},{{ \mathcal I }}_{7}^{{\rm{max}}},{{ \mathcal I }}_{8}^{{\rm{max}}}\}$, $\{{{ \mathcal I }}_{9}^{{\rm{max}}},{{ \mathcal I }}_{10}^{{\rm{max}}}\}$ share surprisingly many common properties. In fact, it is impossible to know that inequalities ${{ \mathcal I }}_{3}^{{\rm{max}}}$ and ${{ \mathcal I }}_{4}^{{\rm{max}}}$ are inequivalent [36] (i.e., cannot be transformed from one to the other via equation (4) and/or relabeling of parties, inputs and outputs) based on the properties that we have analyzed. Their inequivalence is eventually confirmed by computing their canonical representation [77] via the platform provided in [76].

As depicted in figure 1, our experimental setup [86, 87] can be subdivided into three parts: preparation, manipulation and detection of entangled photons. By pumping a periodically poled KTiOPO₄ (PPKTP) crystal with a quasi-monochromatic ND:YVO₄ (Coherent Verdi V5) laser with a central wavelength ${\lambda }_{p,c}=532\;{\rm{nm}}$, collinearly phase-matched type-0 spontaneous parametric down-conversion (SPDC) is exploited. Up to first-order in perturbation theory, the corresponding biphoton state is described by

$$\begin{eqnarray}&&| \psi \rangle ={\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}\omega \;{\rm{\Lambda }}(\omega )\;{\hat{a}}_{i}^{\dagger }(\omega )\;{\hat{a}}_{s}^{\dagger }(-\omega )| 0{\rangle }_{i}| 0{\rangle }_{s},\end{eqnarray} \tag{ 14 }$$

where the leading order vacuum state is omitted. By acting on the composite vacuum state $| 0{\rangle }_{i}| 0{\rangle }_{s}$, the operators ${\hat{a}}_{i,s}^{\dagger }(\omega )$ create the idler (i) and signal (s) photon at relative frequency ω (with respect to $\frac{{\omega }_{p,c}}{2}$). Since we assume a continuous pump field as well as degenerate center frequencies ${\omega }_{i,c}={\omega }_{s,c}=\frac{{\omega }_{p,c}}{2}$, for idler and signal photon, respectively, the joint spectral amplitude (JSA), in general being a two-dimensional function denoted by ${\rm{\Lambda }}({\omega }_{i},{\omega }_{s})$, simplifies to ${\rm{\Lambda }}(\omega )$ with $\omega \equiv {\omega }_{s}=-{\omega }_{i}$, as used in equation (14).

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The subsequent manipulation part is realized via a prism-based pulse shaping configuration including a spatial light modulator (SLM, Jenoptik, SLM-S640d) as a reconfigurable modulation device. The SLM is endowed with two nematic liquid crystal displays and is used in transmission mode. The respective effect on each photon spectrum can be described by a complex transfer function ${M}^{i,s}(\omega )$ which transforms the JSA according to

$$\begin{eqnarray}&&\tilde{{\rm{\Lambda }}}(\omega )={\rm{\Lambda }}(\omega )M(\omega ),\end{eqnarray} \tag{ 15 }$$

where $M(\omega )={M}^{i}(\omega ){M}^{s}(-\omega )$. Finally, coincidences between entangled photons are measured by sum-frequency generation (SFG) using a second PPKTP crystal. The resulting up-converted photons are then imaged onto the photosensitive area of a single photon counting module (SPCM, ID Quantique id100-20-uln). Accordingly, the signal detected by the latter is described by the first-order coherence function

$$\begin{eqnarray}&&S\propto {\left|{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}\omega M(\omega ){\rm{\Gamma }}(\omega )\right|}^{2},\end{eqnarray} \tag{ 16 }$$

where the JSA is modified by the acceptance bandwidth of the detection crystal ${\rm{\Phi }}(\omega )$ such that ${\rm{\Gamma }}(\omega )\propto {\rm{\Lambda }}(\omega ){\rm{\Phi }}(\omega )$. To incorporate the finite spectral resolution at the SLM plane we further convolute the JSA with a Gaussian modeled point-spread function (PSF) denoted with ${{\rm{\Upsilon }}}_{\mathrm{PSF}}(\omega )$ according to

$$\begin{eqnarray}&&{\rm{\Gamma }}(\omega )\to {{\rm{\Gamma }}}_{\mathrm{PSF}}(\omega )\propto ({\rm{\Gamma }}\otimes {{\rm{\Upsilon }}}_{\mathrm{PSF}})(\omega ).\end{eqnarray} \tag{ 17 }$$

To access entangled two-qudit states we project the continuous biphoton state of equation (14) onto a d²-dimensional discrete subspace spanned by orthonormal basis states $| j{\rangle }_{i}| k{\rangle }_{s}$, where $| j{\rangle }_{i,s}\equiv {\displaystyle \int }_{-\infty }^{\infty }{f}_{j}^{i,s}(\omega ){\hat{a}}_{i,s}^{\dagger }(\omega )| 0{\rangle }_{i,s}$ (see [55] for details). Here, the corresponding basis functions ${f}_{j}^{i,s}(\omega )$ are chosen according to a frequency-bin pattern which is given by

$$\begin{eqnarray}{f}_{j}^{i,s}(\omega )=\left\{\begin{array}{cl}1/\sqrt{{\rm{\Delta }}{\omega }_{j}} & \quad \mathrm{for}\quad | \omega -{\omega }_{j}| \lt {\rm{\Delta }}{\omega }_{j}/2,\\ 0 & \quad \mathrm{otherwise}.\end{array}\right.\end{eqnarray} \tag{ 18 }$$

If we impose that adjacent bins do not overlap, the (normalized) projected state can be rewritten in the form of a discrete two-qudit state

$$\begin{eqnarray}&&| {\psi }_{d}\rangle =\displaystyle \sum _{j=0}^{d-1}\;{c}_{j}\;| j{\rangle }_{i}| j{\rangle }_{s}.\end{eqnarray} \tag{ 19 }$$

For d = 3 the corresponding frequency-bin pattern, together with the associated basis states, is shown exemplarily in figure 1. We further, decompose the transfer functions ${M}^{i,s}(\omega )$ in terms of the basis functions ${f}_{j}^{i,s}(\omega )$ such that

$$\begin{eqnarray}&&{M}^{i,s}(\omega )=\displaystyle \sum _{j=0}^{d-1}\;{u}_{j}^{i,s}\;{f}_{j}^{i,s}(\omega )=\displaystyle \sum _{j=0}^{d-1}\;| {u}_{j}^{i,s}| \;{{\rm{e}}}^{{\rm{i}}{\phi }_{j}^{i,s}}\;{f}_{j}^{i,s}(\omega ).\end{eqnarray} \tag{ 20 }$$

Experimentally, the SLM consists of two liquid–crystal displays. This allows us to manipulate the spectral phase ${\phi }_{j}^{i,s}$ and amplitude $| {u}_{j}^{i,s}|$ of the signal and idler photon independently [88]. The latter is realized by an SLM induced polarization modulation on the entangled photons and an SFG crystal which only up-converts h-polarized photons. The combination of the SLM and the SFG coincidence detection finally realizes a projective measurement of $| {\psi }_{d}\rangle$ onto a direct product state $| \chi \rangle$ to be defined below. Accordingly, the discretized expression of equation (16) reads

$$\begin{eqnarray}&&{S}^{(d)}\propto {\left|\displaystyle \sum _{j=0}^{d-1}{u}_{j}^{i}{u}_{j}^{s}{c}_{j}\right|}^{2}={| \langle \chi | {\psi }_{d}\rangle | }^{2}.\end{eqnarray} \tag{ 21 }$$

To be consistent with the framework of a Bell scenario we identify from now on the idler photon with Alice $(A\leftrightarrow i)$ and the signal photon with Bob $(B\leftrightarrow s)$. The set of correlations $P(a,b| x,y)$ is then related to ${S}^{(d)}$ according to $P(a,b| x,y)\propto {S}^{(d)}$ with a state $| \chi \rangle$ given by

$$\begin{eqnarray}| {\chi }_{a| x,b| y}\rangle & = & | a{\rangle }_{A}^{x}| b{\rangle }_{B}^{y}\\ & = & \left(\displaystyle \sum _{j=0}^{d-1}{u}_{j}^{x}(\{{\eta }_{k}^{x}\})| j{\rangle }_{A}\right)\left(\displaystyle \sum _{{j}^{\prime }=0}^{d-1}{u}_{{j}^{\prime }}^{y}(\{{\theta }_{k}^{y}\})| {j}^{\prime }{\rangle }_{B}\right),\end{eqnarray} \tag{ 22 }$$

where ${M}_{a| x}\otimes {M}_{b| y}=| a\rangle {}_{A}^{x}{}_{A}^{x}\langle a| \otimes | b\rangle {}_{B}^{y}{}_{B}^{y}\langle b|$. Each set of states $\{{| a\rangle }_{A}^{x}\}$, $\{{| b\rangle }_{B}^{y}\}$ in equation (22) represents the most general d-dimensional orthonormal basis according to SU(d) and the optimal measurement settings $\{{\eta }_{k}^{x}\}$ and $\{{\theta }_{k}^{y}\}$ with $k\in \{1,...,{d}^{2}-1\}$ are obtained by maximizing the violation of the Bell inequality under consideration. Experimentally, the set of correlations ${\vec{P}}_{{\rm{exp}}}$ is determined by measuring an average count rate $N\propto {S}^{(d)}$ over a certain time T. In particular, under the assumption that the source and measurements are both independent and identically distributed (i.i.d.), the observed correlation components ${P}_{{\rm{exp}}}(a,b| x,y)$ are finally estimated via the relative frequencies

$$\begin{eqnarray}&&{P}_{{\rm{exp}}}(a,b| x,y)=\displaystyle \frac{N(a,b,x,y)}{{\displaystyle \sum }_{{a}^{\prime },{b}^{\prime }}N({a}^{\prime },{b}^{\prime },x,y)}\end{eqnarray} \tag{ 23 }$$

for all $a,b,x,y$.

In this section, experimental results obtained for the Bell inequality violation of ${I}_{3}^{+}$, ${{ \mathcal I }}_{12}^{{\rm{max}}}$ and ${{ \mathcal I }}_{14}^{{\rm{max}}}$ are provided. The quantum value for each of these inequalities is calculated using a representation of the corresponding Bell inequality expressed in the form of equation (2). Specifically, to obtain the Bell coefficients ${\alpha }_{{ab}}^{{xy}}$ of ${{ \mathcal I }}_{12}^{{\rm{max}}}$ and ${{ \mathcal I }}_{14}^{{\rm{max}}}$ from the coefficients given in table 1, we apply the following transformation

$$\begin{eqnarray}{\alpha }_{{ab}}^{{xy}} & = & {\beta }_{{ab}}^{{xy}}\quad \mathrm{if}\;a,b\in \{0,1,2\}\;\mathrm{and}\;\;x,y\in \{1,2\},\\ {\alpha }_{{ab}}^{00} & = & {\beta }_{{ab}}^{00}+{\beta }_{A,a}^{0}+{\beta }_{B,b}^{0}\quad \mathrm{if}\;a,b\in \{0,1,2\},\\ {\alpha }_{{ab}}^{x0} & = & {\beta }_{{ab}}^{x0}+{\beta }_{A,a}^{x}\quad \mathrm{if}\;a,b\in \{0,1,2\}\;\mathrm{and}\;\;x\in \{1,2\},\\ {\alpha }_{{ab}}^{0y} & = & {\beta }_{{ab}}^{0y}+{\beta }_{B,b}^{y}\quad \mathrm{if}\;a,b\in \{0,1,2\}\;\mathrm{and}\;\;y\in \{1,2\},\end{eqnarray} \tag{ 24 }$$

where all coefficients that were not defined in table 1, such as ${\beta }_{22}^{{xy}}$, ${\beta }_{A,2}^{x}$, ${\beta }_{B,2}^{y}$ etc., are understood to be zero in the above equation.

The experimental results—obtained by performing the optimized measurements for each inequality and for each fixed value of γ—are shown in figure 2 as a function of γ. Note that for the experiment on ${I}_{3}^{+}$ and ${{ \mathcal I }}_{14}^{{\rm{max}}}$, the preparation of a maximally entangled two-qutrit state $| {{\rm{\Phi }}}_{3}^{+}\rangle$ and two-qubit state $| {{\rm{\Phi }}}^{+}\rangle$, respectively, are of particular interest. The characteristic shape of a SPDC spectrum (figure 1), however, naturally leads to a non-maximally entangled quantum state, due to unequally distributed probability amplitudes. In our experiments, $| {{\rm{\Phi }}}^{+}\rangle$ and $| {{\rm{\Phi }}}_{3}^{+}\rangle$ are thus prepared by applying the Procrustean method of entanglement concentration [89]. Moreover, for the quantum violation of ${{ \mathcal I }}_{12}^{{\rm{max}}}$, in order to implement the non-projective POVM element, the two-qubit state $| \psi (\gamma ,0)\rangle$ is embedded in a two-qutrit space, i.e., after preparing the state $| {{\rm{\Phi }}}_{3}^{+}\rangle$ as described above, the two Schmidt coefficients c₀ and c₁ are changed according to table 2 while setting c₂ to zero.

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To quantify the imperfection in our setup, we consider the symmetric noise model of equation (8) with ρ being the pure state $| \psi (\gamma ,0)\rangle$ or $| \psi (\gamma ,1)\rangle$ of equation (11) with the appropriate value of γ. From the measured quantum value of each inequality and the theoretically computed maximum quantum violation for each of the aforementioned states, we can then determine the mixing parameter ${\nu }_{d}^{{ \mathcal I }}(\gamma )\in [0,1]$ that gives rise to the observed correlations. The final mixing parameter reported in the caption of figure 2 is the mean value obtained by averaging ${\nu }_{d}^{{ \mathcal I }}(\gamma )$ over different values of γ.

To gain insight into the signaling nature of our experimentally determined correlations, we show in figure 3 the differences

$$\begin{eqnarray}&&{\rm{\Delta }}P(a| x,{y}_{\mathrm{1,2}}):= | P(a| x,{y}_{1})-P(a| x,{y}_{2})| \quad {y}_{1}\ne {y}_{2}\end{eqnarray} \tag{ 25 }$$

for all $a,x,{y}_{1},{y}_{2}\in \{0,1,2\}$ as well as

$$\begin{eqnarray}&&{\rm{\Delta }}P(b| {x}_{\mathrm{1,2}},y):= | P(b| {x}_{1},y)-P(b| {x}_{2},y)| \quad {x}_{1}\ne {x}_{2}\end{eqnarray} \tag{ 26 }$$

for all $b,{x}_{1},{x}_{2},y\in \{0,1,2\}$ derived from ${\vec{P}}_{{\rm{exp}}}$ for the measurement of ${{ \mathcal S }}_{12}$ when $\gamma =1$. In particular, each subplot on the left (blue) shows equation (25) whereas each subplot on the right (red) shows equation (26), thereby indicating the extent of signaling, respectively, from Bob to Alice and from Alice to Bob. If the non-signaling condition is respected but only deviates from equation (4) due to statistical fluctuations, one expects all these differences to vanish within the statistical uncertainty. This is indeed the case for the correlation ${\vec{P}}_{{\rm{exp}}}$ obtained during the measurement of ${{ \mathcal S }}_{12}$ for $\gamma =1$ (figure 3) as well as other values of $\gamma \in [0,1]$ (not shown). Likewise, the signaling nature of correlations ${\vec{P}}_{{\rm{exp}}}$ obtained during the measurement of ${{ \mathcal S }}_{14}$ for $\gamma =1$ (figure 4) as well as other values of $\gamma \in [0,1]$ (not shown) can essentially be understood via statistical fluctuations.

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On the contrary, as can be seen in figure 5, even by considering a statistical uncertainty of two standard deviations, some of the differences in the conditional marginal distributions obtained for $\gamma =1$ during the measurement of ${{ \mathcal S }}_{0}$ (i.e., for inequality ${I}_{3}^{+}$) do not vanish. As such, it is unlikely that the signaling observed in these correlations can be accounted for entirely through statistical fluctuations. Indeed, as separate numerical simulations indicate, the amount of signaling in these data collected for the qutrit measurement is more a consequence of the imaging arrangement. Due to the existing PSF, mentioned in section 4.1, the idler and signal photon are not completely spatially separated at the SLM plane. Consequently, the modulation of one photon could influence the other photon, yielding signaling data. Moreover, due to the limited transverse spread of the total spectrum, a stronger overlap of different spectral components naturally occurs since the spacing ${\rm{\Delta }}\omega$ of the frequency-bin pattern, given in equation (18), is smaller for qutrits than for qubits.

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Having established confidence that our measured correlations for ${{ \mathcal I }}_{12}^{{\rm{max}}}$ and ${{ \mathcal I }}_{14}^{{\rm{max}}}$ do not contradict the non-signaling conditions, we can now proceed with further analysis based on these measured correlations. To this end, if we now make the assumption that the underlying true quantum distribution also violates the measured Bell inequality by the same amount, then some simple statements, such as the amount of entanglement present in the quantum states prepared, can already be made based on the extent to which the observed correlation violates the respective Bell inequalities [23]. Not surprisingly, better estimates can often be obtained if, instead, full measurement statistics are employed in the analysis (see, for instance, [91, 92] in the context of randomness evaluation).

The tools that have been developed for such purposes—such as the linear program given in equation (A1), or the semidefinite programs described in [19, 23, 25, 91, 92]—however, require explicitly that the correlation to be analyzed satisfies the non-signaling condition, equation (4). As described above, although the correlations that we obtained are compatible—within statistical uncertainty—with the non-signaling conditions, the raw correlations themselves do not. In order to take advantage of the full probability distribution for subsequent analysis of the source, some form of post-processing of the raw correlation would be necessary. In [92], the authors removed the signaling in the raw correlation by projecting it into the non-signaling subspace whereas in [86], the authors achieved this by computing the non-signaling distribution that is closest to the raw distribution according to the relative entropy. While the former approach works for the correlations analyzed in [92], both these approaches suffer from the immediate risk that the non-signaling correlation derived may not be quantum realizable, see equation (3a). To minimize this risk, we will compute instead the quantum correlation nearest to the experimentally determined correlation.

In practice, however, there are two other technical difficulties that need to be overcome before we can apply existing techniques for further analysis. Firstly, the exact characterization of the set of quantum correlations for any given Bell scenario is not known and appears to be a formidable task (see, e.g., [79, 80]). To overcome this, we shall consider instead a converging hierarchy of relaxations to the set of quantum correlation ${ \mathcal Q }$, such as that discussed in [79, 80] or [23]—each of these relaxations defines a superset to ${ \mathcal Q }$ and in the asymptotic limit, one recovers ${ \mathcal Q }$.

Secondly, to compute the 'nearest' quantum⁷ approximation to the experimentally determined correlation, we need to choose an appropriate metric, i.e., a distance measure and there is apparently no unique choice to this. To facilitate computation, for the present purpose, we shall consider a distance measure based on the ℓ₁-norm (which corresponds to minimizing the least absolute deviation). As shown in appendix C, the problem of determining a correlation ${\vec{P}}_{{{ \mathcal Q }}_{{\ell }}}\in {{ \mathcal Q }}_{{\ell }}$ which minimizes the distance $| | {\vec{P}}_{{{ \mathcal Q }}_{{\ell }}}-{\vec{P}}_{{\rm{exp}}}| {| }_{1}$—with ${{ \mathcal Q }}_{{\ell }}$ being the ℓ-level of the hierarchy defined in [79] or [23]—can then be cast as a semidefinite program, which can be efficiently solved on a computer. In addition, in order to remove possible degeneracies involved in the computation, we also impose the condition that the distribution sought for, ${\vec{P}}_{{{ \mathcal Q }}_{{\ell }}}$, reproduces the quantum value of the inequalities that we measured in the experiment, i.e., $\vec{\beta }\cdot {\vec{P}}_{{{ \mathcal Q }}_{{\ell }}}=\vec{\beta }\cdot {\vec{P}}_{{\rm{exp}}}$.

As an illustration of the above procedures, we show in figure 6 an estimation of the entanglement—measured according to negativity [58]—present in our source computed using the⁸ technique introduced in [23]. Evidently, the negativity lower bound obtained from the (approximated) full data are generally higher than that arising from the quantum violation of the respective Bell inequalities. In fact, as can be seen in figure 6, the negativity estimated from the full measurement data are sometimes even higher than that obtained from the device-dependent estimation assuming a symmetric noise model. Note also that for each of these Bell experiments, the measured correlations for certain small values of γ are not good enough to violate the corresponding Bell inequalities. However, when the full data is taken into account, we are sometimes able to obtain non-trivial estimates of the entanglement⁹ present in the underlying state.

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To test the reliability of the proposed method, we have also computed the negativity (lower bound) by assuming that the true quantum value is the measured quantum value plus (minus) one standard deviation (see dotted (dashed) line in figure 6). As can be seen in the figure, except for one instance, namely for ${{ \mathcal I }}_{14}^{{\rm{max}}}$ and when $\gamma =0$, all negativities estimated using the actual value fit well between those estimated assuming the shifted values. For this exceptional instance, the negativity estimated from the full data (and the assumption that the true quantum value is the measured value less one σ) turns out to give an even larger (and nonzero) value compared with the case when the assumed quantum value is the measured value (or with one standard deviation above). A possible cause for this somewhat unexpected result is that even with an assumption of the true quantum value, there may be more than one ${\vec{P}}_{{{ \mathcal Q }}_{{\ell }}}\in {{ \mathcal Q }}_{{\ell }}$ that minimizes the ℓ₁ distance to the raw correlation, and the solver happens to have returned a ${\vec{P}}_{{{ \mathcal Q }}_{{\ell }}}$ that gives the highest value of negativity (when the assumed quantum value is the smallest of the three cases considered). Another unusual feature associated with this exceptional instance is that, by right, for $\gamma =0$, our system should have produced a separable state and is thus not capable of violating any Bell inequality (nor producing correlation to be associated with non-trivial negativity). Thus, if the estimation that we have obtained using the (approximated) full data (and the assumption that the true quantum value is indeed the measured value minus one σ) is legitimate, then this conclusion obtained from the measured data nicely illustrates how our trust in the measurement and preparation devices may be misguided.